Nonequilibrium Cooperative Phenomena in Physics and Related Fields, 1984 NATO Science Series B: Series, Vol. 116

This volume contains the lectures and invited seminars pre­ sented at the NATO Advanced Study Institute on NON-EQUILIBRIUM COOPERATIVE PHENOMENA IN PHYSICS AND RELATED FIELDS that was held at EL ESCORIAL (MADRID), SPAIN, on August 1-11, 1983. Most nonlinear problems in dissipative systems, i . e . , most mathematical models in SYNERGETICS are highly trans disciplinary in practice and the list of lecturers and participants at the ASI reflects this di versi ty both in background and interest. The presentation of the material fell into two main categories: tutopia~ Zectures on some basic ideas and methods, both experimental and theoretical, intended to lay a common base for all participants, and a series of more specific lectures and seminars, serving the purpose of exemplying selected but typical applications in their current state of development. Topics were chosen for their basic interest as well as for their potential for applications (laser, hydrodynamics, liquid crystals, EHD, combustion, thermoelasticity, etc. ). We had more seminars and some of the oral presentations were supported or complemented with 16 mm films and on occasion with experimental demonstrations including a special seminar, a social one on broken symmetries in Art and Music. There is here no record of these non-standard acti vi ties. We had, indeed, quite a heavy load for which I was fully responsible. However, the reader and, above all, the participants at the ASI ought to be aware of the fact that in Spain, with.

A. Introduction: Synergetics and Non-Equilibrium Phase Transitions.- Synergetics.- 1. Introduction.- 2. Outline of the general approach.- 3. A brief outline of the mathematical approach.- 4. Some simple examples.- 5. Generalized Ginzburg-Landau equations.- 6. Some further applications.- 7. Hierarchies of temporal patterns: from oscillations to chaos.- 8. Outlook.- Phase transition analogies: magnets, lasers and fluid flows.- 1. Introduction.- 2. Laser threshold- Second — order phase transition analogy.- 3. Laser with saturable absorber — First-order phase transition.- 4. From lasers to fluid flows.- B. Lasers and Quantum Optics.- Collective phenomena in Quantum Optics.- 1. Physics of stimulated emission processes.- 2. What is quantum and what is classical in laser physics.- 3. Bifurcations in non-equilibrium systems.- 4. Photon statistics and the laser threshold.- 5. Turbulence in quantum optics.- Optical Bistability and related topics.- 1. Introduction.- 2. A practical optical bistable device: Recent advances.- 3. Theory of optical bistability in a ring cavity.- 3.1 Steady-state behavior in absorptive optical bistability with zero cavity detuning.- 3.2 The effect of dispersion — Kerr medium.- 4. The mean field model of optical bistability.- 5. Self-pulsing and chaotic behavior.- 6. Instability modes.- 6.1 Resonant-mode instability.- 6.2 Off-resonance mode instability.- Laser with intracavity absorber: Q-switching and multistable nonlinear oscillations and chaos.- 1. Introduction.- 2. Phase diagram.- 3. Soft oscillatory lasing.- 4. Q-switching and coexistence between oscillations and cw lasing.- 5. Coexistence of various oscillations including Q-switching.- 6. A Lorenz-like strange (aperiodic) solution and its fractal dimension.- 7. A Feigenbaum period-doubling cascade, the subsequent strange (aperiodic) attractor and its fractal dimension.- Non-linear operation of CO2 lasers with intracavity saturable absorbers.- Experimental apparatus.- Experimental results.- Theoretical model.- Conclusion.- Structurally stable bifurcations in optical bistability.- 1. Introduction.- 2. Self-pulsing and optical bistability.- 3. Polarization switching.- C. Action of Intense Laser Fields.- Allowed nuclear beta decay in an intense laser field.- Wavefunctions.- Transition rate.- Alternative derivations of the quasiclassical Transition ratel.- Electron distributions.- Nuclear lifetime.- The nonclassical regime.- Summary.- D. Instabilities and Convection in Liquids and Liquid Crystals.- Thermohydrodynamic instabilities: Buoyancy-thermocapillary convection.- 1. Introduction.- 2. Interfacial convection: heuristic arguments.- 3. Buoyancy — thermocapillary equations.- 4. Necessary conditions for the onset of convection.- 5. Sufficient conditions for the onset of convection.- 6. Comments.- Thermohydrodynamic instability in nematic liquid crystals: a summary of arguments and conditions for some simple geometries.- 1. Introduction.- 2. Rayleigh-Benard convection.- 3. Rotating annulus convection with radial temperature gradients.- Neutron scattering studies of phase transitions in equilibrium and nonequilibrium systems.- 1. Introduction.- 1a. Neutron scattering.- 1b. Phase transition in equilibrium.- 2. Neutron scattering studies of nonequilibrium phase transitions.- Electrothermal instabilities in dielectric liquids.- 1. Introduction.- 2. Electrothermal equations.- 3. The onset of steady convection in a simple approach.- 4. Convective flow and the phase transition picture.- 5. Overstability.- Electrohydrodynamic instabilities in nematic with homeotropic boundaries.- EHD convention flow in samples with homeotropic boundaries.- Experiments and results.- E. Convection Diffusion and Reaction.- Gyrotactic buoyant convection and spontaneous pattern formation in algal cell cultures.- Gyrotaxis and focusing.- Convection pattern formation.- Cooperative effects in the time domain.- Other organisms and conditions.- Discussion.- Applications.- Experimental investigations of precipitation patterns.- 1. Introduction.- 2. Spatial and temporal sequence of events in a Liesegand system.- 3. Pattern formation in the presence of low concentration gradients.- 4. Pattern formation in initially uniform colloids.- 5. Complex patterns in periodic precipitation processes.- 6. Conclusion.- F. Combustion.- Optical diagnosis in flows — Applications — Experiments in Combustion.- A powerful non invasive tool: the interaction of light with matter.- Some applications of Rayleigh and Mie scatterings in flow studies.- Experimental study of the flow-flame interaction.- Theory of gaseous combustion.- I. Basic considerations.- I.1 Two feed-back mechanisms.- I.2 The conservation equations.- I.3 The adiabatic temperature of combustion.- I.4 The two different kinds of combustion processes.- I.5 The large activation energy.- II. Premixed flames.- II.1 Position of the problem.- II.2 Existence and unicity.- II.3 The asymptotic expansion.- II.4 Dynamics of flame front.- III. Diffusion flames.- III.1 Position of the problems.- III.2 Ignition regime.- III.3 Diffusion controlled regime.- III.4 Extinction regime.- Bifurcation in Heterogeneous Combustion.- 1. Introduction.- 2. Mathematical model.- 3. Bifurcation diagrams.- 4. Numerical solution.- 5. Conclusions.- G. Instabilities and Nonlinear Phenomena in Solids.- Thermoelasticity and mechanical instabilities.- and Summary.- I. The thermoelastic effect.- II. The thermoelastic-plastic transition in metals.- II.1 Introduction.- II.2 The yield point as the critical point of a dynamical instability.- II.3 “Thermal emission” in 100 Cr6 steel.- II.4 Conclusions.- Non-Equilibrium effects seen in molecular dynamics calculations of Shock waves in Solids.- Discussion.- Conclusions.- H. Deterministic (Continuous and Discrete) Mathematics of Nonlinear Problems.- Current topics in reaction-diffusion systems.- 1. Introduction.- 2. The simplest wave fronts.- 3. Slowly varying fronts.- 4. Coupling with another reactant; propagator-controller systems.- 5. Accounting for dissusion of V.- 6. Target patterns for the Belousov-Zhabotinsky reagent.- 7. The generation of spirals.- 8. Compound layers and stationary solutions.- 9. Small wave trains and associated solutions.- 10. Piecing together a global picture for a model problem.- Discrete nonlinear dynamics.- 1. Introduction.- 2. Why discrete dynamics?.- 3. Experiments.- 4. Trifurcation?.- 5. Density of states, renormalization, and chaos map.- 6. Random numbers?.- 7. Period-doubling renormalization group.- 8. Deterministic Brownian motion?.- 9. Windows of nondiffusive states.- 10. Anomalous diffusion?.- 11. Ornstein-Uhlenbeck process.- Deterministic diffusion — A quality of Chaos.- 1. Introduction.- 2. The onset of diffusion.- 2.1 An example.- 2.2 Master equation.- 2.3 Critical properties of the diffusion coefficient.- 3. Excess noise for intermittent diffusion.- 4. Anomalous diffusion.- 5. Diffusion in two dimensions.- I. Stochastic Description of Non Linear Problems.- Stochastic space-time problems.- 1. Reaction-diffusion models.- 1.1 Introduction and motivation.- 1.2 Deterministic model. Sobolev spaces.- 1.3 Markov jump processes.- 1.4 Stochastic model of reaction with diffusion.- 1.5 Consistency of the stochastic and deterministic model (Thermodynamic limit).- 1.6 Consistency of the stochastic and deterministic model (Continued ).- 2. Stochastic partial differential equations.- 2.1 Motivation.- 2.2 Deterministic evolution equations. Semigroups.- 2.3 Wiener process and stochastic integrals in Hilbert space.- 2.4 Stochastic evolution equations.- 3. SPDE in reaction-diffusion problems.- 3.1 Introduction. Van Kämpen’s approximation.- 3.2 Central limit theorem.- Stochastic theory of transition phenomena in nonequilibrium systems.- 1. Introduction.- 1A. General formulation.- 1B. Simple models.- 1C. The importance of fluctuations.- 2. Stochastic formulation.- 2A. The master equation.- 2B. Birth and death processes.- 2C. Spatially distributed systems.- 2D. Some important limits.- 2E. Stochastic thermodynamics.- 3. Primary bifurcation.- 3A. Critical behavior.- 3B. Nucleation.- 4. The onset of spatial correlations.- 4A. Reaction-Diffusion systems.- 4B. Heat conduction in nonequilibrium.- 5. Transient phenomena.- Non-equilibrium systems with random control paramenters.- External noise.- Mathematical description.- Non-white noise triggered oscillations in nonlinear chemical process.- 1. Introduction.- 2. Deterministic model.- 3. Stochastic model.- 4. Numerical results.- 5. Discussion and conclusions.- About some simple Fokker-Planck models.- A very simple model.- Expansion in the nonlinearity parameter.- Some comments.- Dynamics of symmetry breaking: phase coherence in finite and random system.- 1. Spherical limit of TDGL.- 2. Size and dimensionality effects.- 3. Dynamics of the random field instability.- 4. Conclusion.- Pictures of participants.- Participants.- Author Index.